von Neumann. I've read a huge stack of math books, and I have an even bigger stack of unread math books. And it's starting to come together.I've been working for the past 15 months on repairing my rusty math skills, ever since I read a biography of Johnny

自从我读了Johnny von Neumann的传记,我已经为弥补我糟糕的数学技能花了15个月了。读了大量的数学书籍,不过呢,似乎我还有更多没有读。当然我会接着做的。

Let me tell you about it. 现在我就来告诉你这些。

Conventional Wisdom Doesn't Add Up 告别传统观念

First: programmers don't think they need to know math. I hear that so often; I hardly know anyone who disagrees. Even programmers who were math majors tell me they don't really use math all that much! They say it's better to know about design patterns, object-oriented methodologies, software tools, interface design, stuff like that.


And you know what? They're absolutely right. You can be a good, solid, professional programmer without knowing much math.


But hey, you don't really need to know how to program, either. Let's face it: there are a lot of professional programmers out there who realize they're not very good at it, and they still find ways to contribute.


If you're suddenly feeling out of your depth, and everyone appears to be running circles around you, what are your options? Well, you might discover you're good at project management, or people management, or UI design, or technical writing, or system administration, any number of other important things that "programmers" aren't necessarily any good at. You'll start filling those niches (because there's always more work to do), and as soon as you find something you're good at, you'll probably migrate towards doing it full-time.


In fact, I don't think you need to know anything, as long as you can stay alive somehow.


So they're right: you don't need to know math, and you can get by for your entire life just fine without it.


But a few things I've learned recently might surprise you:


Math is a lot easier to pick up after you know how to program. In fact, if you're a halfway decent programmer, you'll find it's almost a snap.


They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer.


Knowing even a little of the right kinds of math can enable you do write some pretty interesting programs that would otherwise be too hard. In other words, math is something you can pick up a little at a time, whenever you have free time.


Nobody knows all of math, not even the best mathematicians. The field is constantly expanding, as people invent new formalisms to solve their own problems. And with any given math problem, just like in programming, there's more than one way to do it. You can pick the one you like best.


Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)


The Math You Learned (And Forgot) 你学到的数学(和你忘了的数学)

Here's the math I learned in school, as far as I can remember:


Grade School: Numbers, Counting, Arithmetic, Pre-Algebra ("story problems")


High School: Algebra, Geometry, Advanced Algebra, Trigonometry, Pre-Calculus (conics and limits)

高中:代数,几何,高等代数,三角学,微积分先修课 (二次曲线论和极限)

College: Differential and Integral Calculus, Differential Equations, Linear Algebra, Probability and Statistics, Discrete Math


How'd they come up with that particular list for high school, anyway? It's more or less the same courses in most U.S. high schools. I think it's very similar in other countries, too, except that their students have finished the list by the time they're nine years old. (Americans really kick butt at monster-truck competitions, though, so it's not a total loss.)


Algebra? Sure. No question. You need that. And a basic understanding of Cartesian geometry, too. Those are useful, and you can learn everything you need to know in a few months, give or take. But the rest of them? I think an introduction to the basics might be useful, but spending a whole semester or year on them seems ridiculous.

代数?是的。没问题。你需要代数。和一些理解解析几何的知识。那些很有用,并且在以后 几个月里,你能学到一切你想要的,十拿九稳的。剩下的呢?我认为一个基本的介绍可能会有用,但是在这上面花整个学期或一年就显得很荒谬了。

I'm guessing the list was designed to prepare students for science and engineering professions. The math courses they teach in and high school don't help ready you for a career in programming, and the simple fact is that the number of programming jobs is rapidly outpacing the demand for all other engineering roles.


And even if you're planning on being a scientist or an engineer, I've found it's much easier to learn and appreciate geometry and trig after you understand what exactly math is — where it came from, where it's going, what it's for. No need to dive right into memorizing geometric proofs and trigonometric identities. But that's exactly what high schools have you do.

即使你打算当一名科学家或者一名工程师,在你理解了什么是数学之后-- 数学它如何而来,如何而去,为何而生,我发现这更加容易去学习和欣赏几何学和三角学。不必去专研记住几何上的证明和三角恒等式,虽然那确实是高中学校要求你必须去做的。

So the list's no good anymore. Schools are teaching us the wrong math, and they're teaching it the wrong way. It's no wonder programmers think they don't need any math: most of the math we learned isn't helping us.


The Math They Didn't Teach You 他们没有教给你的那部分数学

The math computer scientists use regularly, in real life, has very little overlap with the list above. For one thing, most of the math you learn in grade school and high school is continuous: that is, math on the real numbers. For computer scientists, 95% or more of the interesting math is discrete: i.e., math on the integers.

在现实中,计算机科学家经常使用的数学,跟上面所列的数学仅有很小的重叠。 举个例子,你在中学里学的大部分数学是连续性的:也就是说,那是作为实数的数学。而对于计算机科学家来说,他们所感兴趣的95%也许更多的是离散性的:比如,关于整数的数学。

I'm going to talk in a future blog about some key differences between computer science, software engineering, programming, hacking, and other oft-confused disciplines. I got the basic framework for these (upcoming) insights in no small part from Richard Gabriel's Patterns Of Software, so if you absolutely can't wait, go read that. It's a good book.

我打算在以后的博客中再谈一些有关计算机科学,软件工程,编程,搞些有趣的东东,和其他常常令人犯晕的训练。我已经从Richard Gabriel的 软件的模式 这本书中洞察到一个无关巨细的基本框架。如果你明显的等不下去的话,去读吧。是本不错的书。

For now, though, don't let the term "computer scientist" worry you. It sounds intimidating, but math isn't the exclusive purview of computer scientists; you can learn it all by yourself as a closet hacker, and be just as good (or better) at it than they are. Your background as a programmer will help keep you focused on the practical side of things.


The math we use for modeling computational problems is, by and large, math on discrete integers. This is a generalization. If you're with me on today's blog, you'll be studying a little more math from now on than you were planning to before today, and you'll discover places where the generalization isn't true. But by then, a short time from now, you'll be confident enough to ignore all this and teach yourself math the way you want to learn it.


For programmers, the most useful branch of discrete math is probability theory. It's the first thing they should teach you after arithmetic, in grade school. What's probability theory, you ask? Why, it's counting. How many ways are there to make a Full House in poker? Or a Royal Flush? Whenever you think of a question that starts with "how many ways..." or "what are the odds...", it's a probability question. And as it happens (what are the odds?), it all just turns out to be "simple" counting. It starts with flipping a coin and goes from there. It's definitely the first thing they should teach you in grade school after you learn Basic Calculator Usage.

对程序员来说,最有效的离散数学的分支是概率理论。这是你在学校学完基本算术后的紧接着的课。你会问,什么是概率理论呢?你就数啊,看有多少次出现满堂彩?或者有多次是同花顺。 不管你思考什么问题如果是以"多少种途径。。。"或"有多大几率的。。。",那就是离散问题。当他发生时,都 转化成"简单"的计数。抛个硬币看看。。。? 毫无疑问在他们教你基本的计算用法后他们会教你概率理论。

I still have my discrete math textbook from college. It's a bit heavyweight for a third-grader (maybe), but it does cover a lot of the math we use in "everyday" computer science and computer engineering.


Oddly enough, my professor didn't tell me what it was for. Or I didn't hear. Or something. So I didn't pay very close attention: just enough to pass the course and forget this hateful topic forever, because I didn't think it had anything to do with programming. That happened in quite a few of my comp sci courses in college, maybe as many as 25% of them. Poor me! I had to figure out what was important on my own, later, the hard way.


I think it would be nice if every math course spent a full week just introducing you to the subject, in the most fun way possible, so you know why the heck you're learning it. Heck, that's probably true for every course.


Aside from probability and discrete math, there are a few other branches of mathematics that are potentially quite useful to programmers, and they usually don't teach them in school, unless you're a math minor. This list includes:


Statistics, some of which is covered in my discrete math book, but it's really a discipline of its own. A pretty important one, too, but hopefully it needs no introduction.


Algebra and Linear Algebra (i.e., matrices). They should teach Linear Algebra immediately after algebra. It's pretty easy, and it's amazingly useful in all sorts of domains, including machine learning.


Mathematical Logic. I have a really cool totally unreadable book on the subject by Stephen Kleene, the inventor of the Kleene closure and, as far as I know, Kleenex. Don't read that one. I swear I've tried 20 times, and never made it past chapter 2. If anyone has a recommendation for a better introduction to this field, please post a comment. It's obviously important stuff, though.

数理逻辑。我有相当完整的关于这门学科的书没有读,是Stephen Kleene写的,克林闭包的发明者,我所知道的还有就是Kleenex。这个就不要读了。我发誓我已经尝试了不下20次,却从没有读完第二章。如果哪位牛掰有什么更好的入门建议的话可以给我推荐。虽然,这明显是非常重要的一部分。

Information Theory and Kolmogorov Complexity. Weird, eh? I bet none of your high schools taught either of those. They're both pretty new. Information theory is (veeery roughly) about data compression, and Kolmogorov Complexity is (also roughly) about algorithmic complexity. I.e., how small you can you make it, how long will it take, how elegant can the program or data structure be, things like that. They're both fun, interesting and useful.


There are others, of course, and some of the fields overlap. But it just goes to show: the math that you'll find useful is pretty different from the math your school thought would be useful.


What about calculus? Everyone teaches it, so it must be important, right?


Well, calculus is actually pretty easy. Before I learned it, it sounded like one of the hardest things in the universe, right up there with quantum mechanics. Quantum mechanics is still beyond me, but calculus is nothing. After I realized programmers can learn math quickly, I picked up my Calculus textbook and got through the entire thing in about a month, reading for an hour an evening.


Calculus is all about continuums — rates of change, areas under curves, volumes of solids. Useful stuff, but the exact details involve a lot of memorization and a lot of tedium that you don't normally need as a programmer. It's better to know the overall concepts and techniques, and go look up the details when you need them.

微积分都是关于连续统的 -- 变化的比率, 曲线的面积, 立体的体积。是些有用的东西,但是实际细节却包含大量的记忆量并且枯燥,作为一个程序员来说根本不需要这些。 更好的方法是从整体上了解那些概念和技术,在必要的时候再去查询那些细节。

Geometry, trigonometry, differentiation, integration, conic sections, differential equations, and their multidimensional and multivariate versions — these all have important applications. It's just that you don't need to know them right this second. So it probably wasn't a great idea to make you spend years and years doing proofs and exercises with them, was it? If you're going to spend that much time studying math, it ought to be on topics that will remain relevant to you for life.

几何,三角,微分,积分,圆锥曲线,微分方程,和他们的多维和多元 -- 这些都有重要的应用。不过这时候不需要你去了解它们。这大概不是个好注意让你年复一年的去做证明和它们的练习题,不是吗?如果你打算花大量的时间去学习数学,那也是和你生活相关的部分。

The Right Way To Learn Math 学习数学的正确方法

The right way to learn math is breadth-first, not depth-first. You need to survey the space, learn the names of things, figure out what's what.


To put this in perspective, think about long division. Raise your hand if you can do long division on paper, right now. Hands? Anyone? I didn't think so.


I went back and looked at the long-division algorithm they teach in grade school, and damn if it isn't annoyingly complicated. It's deterministic, sure, but you never have to do it by hand, because it's easier to find a calculator, even if you're stuck on a desert island without electricity. You'll still have a calculator in your watch, or your dental filling, or something,


Why do they even teach it to you? Why do we feel vaguely guilty if we can't remember how to do it? It's not as if we need to know it anymore. And besides, if your life were on the line, you know you could perform long division of any arbitrarily large numbers. Imagine you're imprisoned in some slimy 3rd-world dungeon, and the dictator there won't let you out until you've computed 219308862/103503391. How would you do it? Well, easy. You'd start subtracting the denominator from the numerator, keeping a counter, until you couldn't subtract it anymore, and that'd be the remainder. If pressed, you could figure out a way to continue using repeated subtraction to estimate the remainder as decimal number (in this case, 0.1185678219, or so my Emacs M-x calc tells me. Close enough!)

为什么他们还教你这些呢?为什么我们感到含混心虚讷,如果我们不能记住怎样去做?这不是好像我们需要再次知道她。除此以外,如果你命悬一线,你可以运用任意大的数来做长除法。相象你被囚禁在第三世界的地牢里,那儿的独裁者是不会放你出来的,除非你计算出219308862/103503391。你会怎么做呢?好吧,很容易。你开始从分子减去分母,直到不能再减只剩余数为止。若实在有压力,你可以想个办法,继续使用反复减,估算作为十进制的余数(这种情况下,0。1185678219,Emacs M-x calc 告诉我的。够精确了! )

You could figure it out because you know that division is just repeated subtraction. The intuitive notion of division is deeply ingrained now.


The right way to learn math is to ignore the actual algorithms and proofs, for the most part, and to start by learning a little bit about all the techniques: their names, what they're useful for, approximately how they're computed, how long they've been around, (sometimes) who invented them, what their limitations are, and what they're related to. Think of it as a Liberal Arts degree in mathematics.

学习数学的正确方法是忽略实际的算法和证明,对于大部分情况来说, 。。。:他们的名字,他们的作用,他们计算的大致步骤, (有时是)谁发明了他们,发明了多久了,他们的缺陷是什么,和他们相关的有什么。把数学当文科来学。

Why? Because the first step to applying mathematics is problem identification. If you have a problem to solve, and you have no idea where to start, it could take you a long time to figure it out. But if you know it's a differentiation problem, or a convex optimization problem, or a boolean logic problem, then you at least know where to start looking for the solution.

为什么呢?因为第一步反应在数学上的是问题的确定。如果你有一个问题去解决,并且假设你没有头绪如何开始, 这将花费你很长的时间来弄明白。但如果你知道这是个变异的问题,或者是一个凸优化问题,或者一个布尔的逻辑问题,然后你起码能知道从哪着手开始寻找解决方案。

There are lots and lots of mathematical techniques and entire sub-disciplines out there now. If you don't know what combinatorics is, not even the first clue, then you're not very likely to be able to recognize problems for which the solution is found in combinatorics, are you?

现在有许许多多的数学技术和整个的学科分支。如果你不知道组合逻辑是什么,甚至连听都没听说过, 那么你是不可能意识到在组合逻辑中可以找到的解决答案的问题的,难道不是么?

But that's actually great news, because it's easier to read about the field and learn the names of everything than it is to learn the actual algorithms and methods for modeling and computing the results. In school they teach you the Chain Rule, and you can memorize the formula and apply it on exams, but how many students really know what it "means"? So they're not going to be able to know to apply the formula when they run across a chain-rule problem in the wild. Ironically, it's easier to know what it is than to memorize and apply the formula. The chain rule is just how to take the derivative of "chained" functions — meaning, function x() calls function g(), and you want the derivative of x(g()). Well, programmers know all about functions; we use them every day, so it's much easier to imagine the problem now than it was back in school.

但那实在是个大新闻哪,因为阅读这些领域,学习实际算法,建模和计算结果的方法,记住这些名字都是容易的。在学校里他们教你链式法则,你也能回忆起他们并能运用在考试题上,但有多少学生能真正的了解他们到底意味着什么呢? 所以当他们遇到变种的链式问题时,他们就不懂得如何运用公式了。让人感到讽刺的是,了解这是什么比记住如何运用公式更为容易。链式法则仅仅是如何对链式函数求导的意思,函数 x() 引用函数 g() ,你要求导 x(g()) 。好了,程序员知道所有这些函数相关的;我们每天都使用他们,所以现在比过去在学校更加容易能够想象到问题所在。

Which is why I think they're teaching math wrong. They're doing it wrong in several ways. They're focusing on specializations that aren't proving empirically to be useful to most high-school graduates, and they're teaching those specializations backwards. You should learn how to count, and how to program, before you learn how to take derivatives and perform integration.

这就是为什么我认为他们以错误的方式在教数学。 对大多数高中毕业生来说,他们专门教授的内容,不是可以靠经验来证明数学是如何如何有用的,他们教的那些恰恰是非经验式的内容。在你学习如何求导和做积分之前,你将要学习如何计数,怎样编程。

I think the best way to start learning math is to spend 15 to 30 minutes a day surfing in Wikipedia. It's filled with articles about thousands of little branches of mathematics. You start with pretty much any article that seems interesting (e.g. String theory, say, or the Fourier transform, or Tensors, anything that strikes your fancy.) Start reading. If there's something you don't understand, click the link and read about it. Do this recursively until you get bored or tired.

我认为学习数学最好的方法是每天花15到30分钟逛维基百科。那上面有数千数学分支的相关文章。 可以从一些你感兴趣的文章着手(比如,弦理论,或者,傅立叶变换,或者张量理论,就是能冲击你相象力的东西) 阅读。如果有什么你不理解的,就去了解那些链接。如此这般直到你累到不行为止。

Doing this will give you amazing perspective on mathematics, after a few months. You'll start seeing patterns — for instance, it seems that just about every branch of mathematics that involves a single variable has a more complicated multivariate version, and the multivariate version is almost always represented by matrices of linear equations. At least for applied math. So Linear Algebra will gradually bump its way up your list, until you feel compelled to learn how it actually works, and you'll download a PDF or buy a book, and you'll figure out enough to make you happy for a while.


With the Wikipedia approach, you'll also quickly find your way to the Foundations of Mathematics, the Rome to which all math roads lead. Math is almost always about formalizing our "common sense" about some domain, so that we can deduce and/or prove new things about that domain. Metamathematics is the fascinating study of what the limits are on math itself: the intrinsic capabilities of our formal models, proofs, axiomatic systems, and representations of rules, information, and computation.


One great thing that soon falls by the wayside is notation. Mathematical notation is the biggest turn-off to outsiders. Even if you're familiar with summations, integrals, polynomials, exponents, etc., if you see a thick nest of them your inclination is probably to skip right over that sucker as one atomic operation.


However, by surveying math, trying to figure out what problems people have been trying to solve (and which of these might actually prove useful to you someday), you'll start seeing patterns in the notation, and it'll stop being so alien-looking. For instance, a summation sign (capital-sigma) or product sign (capital-pi) will look scary at first, even if you know the basics. But if you're a programmer, you'll soon realize it's just a loop: one that sums values, one that multiplies them. Integration is just a summation over a continuous section of a curve, so that won't stay scary for very long, either.

然而,从观察数学来说,尝试着明白人们正在试图解决的问题(那些已被证明了的问题某天也许会对你有实际用途), 你会开始在符号中看到相同的类型,你也不再排斥他们。比如,累加符号(大写符号-西格马)或者π(大写符号-pi,连乘符号)起初看上去让人心里没底,即时你了解了他们的基本原理。但如果你是个程序员,你会认识到他仅仅是个循环:一个累加值,一个累乘。积分是一段连续曲线的相加,所以那不会让你郁闷太久。

Once you're comfortable with the many branches of math, and the many different forms of notation, you're well on your way to knowing a lot of useful math. Because it won't be scary anymore, and next time you see a math problem, it'll jump right out at you. "Hey," you'll think, "I recognize that. That's a multiplication sign!"

一旦你习惯了数学的许多分支,和许多不同的符号的格式,你就走在了解许多数学知识的路上了。因为你不再害怕,你将会发现问题,其实他们会自动跳到你面前。"嗨,"你会思索,"我 了解这个。这是乘法符号!"

And then you should pull out the calculator. It might be a very fancy calculator such as R, Matlab, Mathematica, or a even C library for support vector machines. But almost all useful math is heavily automatable, so you might as well get some automated servants to help you with it.

这样你就能扔掉计算器了。有一个充满相象的计算器比如 R,Matlab,Mathematica,甚或是支持向量机的C语言库。但几乎所有有用的数学都是重型自动机,所以你能够让一切都变的自动化。

When Are Exercises Useful? 练习有啥用处呢?

After a year of doing part-time hobbyist catch-up math, you're going to be able to do a lot more math in your head, even if you never touch a pencil to a paper. For instance, you'll see polynomials all the time, so eventually you'll pick up on the arithmetic of polynomials by osmosis. Same with logarithms, roots, transcendentals, and other fundamental mathematical representations that appear nearly everywhere.

在做了几年的业余数学爱好者之后,你打算做更多的数学,甚至你从没碰过铅笔和纸。比如, 你会一直看到多项式,所以最后你会耳濡目染的做起多项式的运算。同样的,对数,根,超越数,和其他到处出现的基本数学原理。

I'm still getting a feel for how many exercises I want to work through by hand. I'm finding that I like to be able to follow explanations (proofs) using a kind of "plausibility test" — for instance, if I see someone dividing two polynomials, I kinda know what form the result should take, and if their result looks more or less right, then I'll take their word for it. But if I see the explanation doing something that I've never heard of, or that seems wrong or impossible, then I'll dig in some more.


That's a lot like reading programming-language source code, isn't it? You don't need to hand-simulate the entire program state as you read someone's code; if you know what approximate shape the computation will take, you can simply check that their result makes sense. E.g. if the result should be a list, and they're returning a scalar, maybe you should dig in a little more. But normally you can scan source code almost at the speed you'd read English text (sometimes just as fast), and you'll feel confident that you understand the overall shape and that you'll probably spot any truly egregious errors.


I think that's how mathematically-inclined people (mathematicians and hobbyists) read math papers, or any old papers containing a lot of math. They do the same sort of sanity checks you'd do when reading code, but no more, unless they're intent on shooting the author down.


With that said, I still occasionally do math exercises. If something comes up again and again (like algebra and linear algebra), then I'll start doing some exercises to make sure I really understand it.


But I'd stress this: don't let exercises put you off the math. If an exercise (or even a particular article or chapter) is starting to bore you, move on. Jump around as much as you need to. Let your intuition guide you. You'll learn much, much faster doing it that way, and your confidence will grow almost every day.


How Will This Help Me? 这些怎样才能帮到我?

Well, it might not — not right away. Certainly it will improve your logical reasoning ability; it's a bit like doing exercise at the gym, and your overall mental fitness will get better if you're pushing yourself a little every day.


For me, I've noticed that a few domains I've always been interested in (including artificial intelligence, machine learning, natural language processing, and pattern recognition) use a lot of math. And as I've dug in more deeply, I've found that the math they use is no more difficult than the sum total of the math I learned in high school; it's just different math, for the most part. It's not harder. And learning it is enabling me to code (or use in my own code) neural networks, genetic algorithms, bayesian classifiers, clustering algorithms, image matching, and other nifty things that will result in cool applications I can show off to my friends.


And I've gradually gotten to the point where I no longer break out in a cold sweat when someone presents me with an article containing math notation: n-choose-k, differentials, matrices, determinants, infinite series, etc. The notation is actually there to make it easier, but (like programming-language syntax) notation is always a bit tricky and daunting on first contact. Nowadays I can follow it better, and it no longer makes me feel like a plebian when I don't know it. Because I know I can figure it out.


And that's a good thing.


And I'll keep getting better at this. I have lots of years left, and lots of books, and articles. Sometimes I'll spend a whole weekend reading a math book, and sometimes I'll go for weeks without thinking about it even once. But like any hobby, if you simply trust that it will be interesting, and that it'll get easier with time, you can apply it as often or as little as you like and still get value out of it.


Math every day. What a great idea that turned out to be!




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